Given that cot(A/2) = x, which trigonometric expression in terms of A correctly represents the value of x?

Introduction / Context:
In this aptitude question on trigonometry, we are asked to express cot(A/2) in terms of the single angle A. Such half angle identities are very useful when we want to simplify trigonometric expressions or solve equations that involve A/2 but the available information is in terms of A only.

Given Data / Assumptions:

• cot(A/2) = x
• A is an acute angle in a right triangle, so all required trigonometric functions are defined
• We need to write x as an expression in terms of cosA and possibly other basic functions

Concept / Approach:
To solve this problem we use standard half angle identities from trigonometry. For an acute angle A, the half angle identity for tangent is tan(A/2) = √[(1 - cosA)/(1 + cosA)]. The reciprocal of tangent is cotangent, so cot(A/2) is equal to 1 / tan(A/2). Using this relation we can derive a corresponding expression for cot(A/2) in terms of cosA only. This gives us an expression that contains a square root of a rational function of cosine.

Step-by-Step Solution:
Step 1: Recall that tan(A/2) = √[(1 - cosA)/(1 + cosA)] for 0 < A < 180 degrees. Step 2: Since cot(A/2) is the reciprocal of tan(A/2), write cot(A/2) = 1 / tan(A/2). Step 3: Substitute the half angle formula for tan(A/2): cot(A/2) = 1 / √[(1 - cosA)/(1 + cosA)]. Step 4: Invert the fraction inside the square root: 1 / √[(1 - cosA)/(1 + cosA)] = √[(1 + cosA)/(1 - cosA)]. Step 5: Therefore x = cot(A/2) = √[(1 + cosA)/(1 - cosA)].

Verification / Alternative check:
We can verify the result by taking a specific acute angle, for example A = 60 degrees. Then A/2 = 30 degrees. Compute cot(30 degrees) which is √3. Now compute √[(1 + cos60 degrees)/(1 - cos60 degrees)]. Since cos60 degrees = 1/2, we obtain √[(1 + 1/2)/(1 - 1/2)] = √[(3/2)/(1/2)] = √3, which matches cot(30 degrees). This confirms that the identity is correct for at least one test value and supports the general result.

Why Other Options Are Wrong:

• Option B: cosecA - cotA equals tan(A/2), not cot(A/2).
• Option C: √[(1 - cosA)/2] is related to sin(A/2), not cot(A/2).
• Option D: √[(1 + cosA)/2] is related to cos(A/2), not cot(A/2).
• Option E: (1 - cosA)/sinA is another standard form of tan(A/2), not of cot(A/2).

Common Pitfalls:
A very common mistake is to confuse the half angle identities for tan(A/2), sin(A/2) and cos(A/2) and to mix up the positions of 1 + cosA and 1 - cosA inside the square root. Another frequent error is to forget that cotangent is the reciprocal of tangent, which leads students to select the expression for tan(A/2) instead of cot(A/2). Careful use of the reciprocal relation and step by step simplification avoids these errors.

Final Answer:
Thus, the correct expression for x is √[(1 + cosA)/(1 - cosA)].